Introduction
According to Kontsevich’s proposal [Reference KontsevichKon09], Fukaya categories of Stein manifolds can be described as global sections of a constructible cosheaf of dg categories on a possibly singular Lagrangian spine onto which the manifold retracts. Various approaches to this proposal have been developed, see for example [Reference NadlerNad14, Reference BocklandtBoc11, Reference Sibilla, Treumann and ZaslowSTZ14, Reference Haiden, Katzarkov and KontsevichHKK14, Reference NadlerNad15, Reference Dyckerhoff and KapranovDK13, Reference NadlerNad13].
The specific case we focus on in this work is the following. Let
$S$
be a compact connected Riemann surface, possibly with boundary, and let
$M\subset S$
be a finite non-empty subset of marked points, so that the complement
$S\setminus M$
is a Stein manifold. Any spanning graph
$\unicode[STIX]{x1D6E4}$
in
$S\setminus M$
provides a Lagrangian spine. In [Reference Dyckerhoff and KapranovDK13], the language of cyclic 2-Segal spaces was used to realize Kontsevich’s proposal in this situation. For every commutative ring
$k$
, this theory produces a constructible cosheaf of
$k$
-linear differential
$\mathbb{Z}/2\mathbb{Z}$
-graded categories on any spanning graph
$\unicode[STIX]{x1D6E4}$
, shows that the dg category
$F(S,M;k)$
of global sections is independent of the chosen graph, and implies a coherent action of the mapping class group of the surface on
$F(S,M;k)$
. It is expected that the resulting dg category is Morita equivalent to a variant of the wrapped Fukaya category of the surface. We refer to
$F(S,M;k)$
as the
$k$
-linear topological Fukaya category of the surface
$(S,M)$
. If the surface
$S\setminus M$
is equipped with a framing then a paracyclic version of the above construction can be used to define a differential
$\mathbb{Z}$
-graded lift of
$F(S,M;k)$
(cf. [Reference LurieLur14, Reference Dyckerhoff and KapranovDK15]). The first part of this paper implements these constructions in the framework of
$\infty$
-categories, which provides the flexibility necessary for our purposes.
A functor
$H$
defined on the category of small
$k$
-linear dg categories with values in a stable
$\infty$
-category
$C$
is called:
-
(1) localizing if
$H$
inverts Morita equivalences and sends exact sequences of dg categories to exact sequences in
$C$
; -
(2)
$\mathbb{A}^{1}$
-homotopy invariant if, for every dg category
$A$
, the functor
$H$
maps
$A\rightarrow A[t]$
to an equivalence in
$C$
.
Examples of localizing
$\mathbb{A}^{1}$
-homotopy invariants are provided by periodic cyclic homology over a field of characteristic
$0$
, homotopy
$K$
-theory,
$K$
-theory with finite coefficients, and topological
$K$
-theory over
$\mathbb{C}$
(cf. [Reference KellerKel98, Reference Tabuada and Van den BerghTVdB15, Reference BlancBla16]). The main result of this work is the following theorem.
Theorem 0.1. Let
$H$
be a localizing
$\mathbb{A}^{1}$
-homotopy invariant with values in a stable
$\infty$
-category
$C$
, and let
$(S,M)$
be a stable marked surface where
$S\setminus M$
is equipped with a framing. Define
$E=H(k)$
to be the object of
$C$
obtained by applying
$H$
to the dg category with one object and endomorphism ring
$k$
. Then there is an equivalence
$$\begin{eqnarray}H(F(S,M;k))\simeq E(S,M)[-1],\end{eqnarray}$$
where the right-hand side denotes the relative homology of the pair
$(S,M)$
with coefficients in
$E[-1]$
.
As a concrete example, we have the following corollary.
Corollary 0.2. Let
$k$
be a field of characteristic
$0$
. We have the formulas
$$\begin{eqnarray}\displaystyle \operatorname{HP}_{0}(F(S,M;k)) & \cong & \displaystyle H_{1}(S,M;k),\nonumber\\ \displaystyle \operatorname{HP}_{1}(F(S,M;k)) & \cong & \displaystyle H_{2}(S,M;k)\nonumber\end{eqnarray}$$
for periodic cyclic homology over
$k$
.
Our proof strategy is as follows: we prove a Mayer–Vietoris type statement for localizing
$\mathbb{A}^{1}$
-homotopy invariants of topological Fukaya categories. Roughly speaking, using the local nature of the topological Fukaya category, the result allows us to reduce the calculation of such an invariant to the case when the surface is a boundary-marked disk. The arguments we use are inspired by methods in Thomason–Trobaugh’s algebraic
$K$
-theory of schemes and may be of independent interest: we establish localization sequences for topological Fukaya categories which are analogous to the ones for derived categories of schemes appearing in [Reference Thomason and TrobaughTT90]. Similar localization techniques for Fukaya categories play a role in [Reference Haiden, Katzarkov and KontsevichHKK14], where the Grothendieck group is computed, and the forthcoming work [Reference Pascaleff and SibillaPS16]. The Mayer–Vietoris theorem expresses
$H(F(S,M))$
as a state sum of a coparacyclic
$1$
-Segal object in a stable
$\infty$
-category. We conclude by proving a delooping statement for such objects, which allows us to compute the state sum explicitly. Finally, I would like to point out that Jacob Lurie has communicated to me a version of Theorem 0.1 that does not assume
$\mathbb{A}^{1}$
-homotopy invariance but assumes that the surface has no internal marked points.
We will use the language of
$\infty$
-categories and refer to [Reference LurieLur09] as a general reference.
1 Cyclic and paracyclic 2-Segal objects
We begin with a translation of some constructions in [Reference Dyckerhoff and KapranovDK13] into the context of
$\infty$
-categories.
1.1 The Segal conditions
We start by formulating the Segal conditions (cf. [Reference SegalSeg74, Reference RezkRez01, Reference Dyckerhoff and KapranovDK12]).
Remark 1.1. Let
$\boldsymbol{\unicode[STIX]{x1D6E5}}$
denote the category of finite non-empty linearly ordered sets. This category contains the simplex category
$\unicode[STIX]{x1D6E5}$
as the full subcategory spanned by the collection of standard ordinals
$\{[n],n\geqslant 0\}$
. For every object of
$\boldsymbol{\unicode[STIX]{x1D6E5}}$
, there exists a unique isomorphism with an object of
$\unicode[STIX]{x1D6E5}$
so that we can identify any diagram in
$\boldsymbol{\unicode[STIX]{x1D6E5}}$
with a unique diagram in
$\unicode[STIX]{x1D6E5}$
. With this in mind, we may use arbitrary finite non-empty linearly ordered sets to describe diagrams in
$\unicode[STIX]{x1D6E5}$
without ambiguity.
Definition 1.2. Let
$C$
be an
$\infty$
-category, and let
$X^{\bullet }:\operatorname{N}(\unicode[STIX]{x1D6E5})\rightarrow C$
be a cosimplicial object in
$C$
.
-
(1) The cosimplicial object
$X^{\bullet }$
is called 1-Segal if, for every
$0<k<n$
, the resulting diagramin
$C$
is a pushout diagram.
-
(2) Let
$P$
be a planar convex polygon with vertices labelled cyclically by the set
$\{0,1,\ldots ,n\}$
,
$n\geqslant 3$
. Consider a diagonal of
$P$
with vertices labelled by
$i<j$
so that we obtain a subdivision of
$P$
into two subpolygons with vertex sets
$\{0,1,\ldots ,i,j,\ldots ,n\}$
and
$\{i,i+1,\ldots ,j\}$
, respectively. The resulting triple of numbers
$0\leqslant i<j\leqslant n$
is called a polygonal subdivision.
-
(i) The cosimplicial object
$X^{\bullet }$
is called 2-Segal if, for every polygonal subdivision
$0\leqslant i<j\leqslant n$
, the resulting diagram(1.3)in
$C$
is a pushout diagram.
-
(ii) We say
$X^{\bullet }$
is a unital
$2$
-Segal object if, in addition to (i), for every
$0\leqslant k<n$
, the diagram(1.4)in
$C$
is a pushout diagram.
Proposition 1.5. Every
$1$
-Segal cosimplicial object
$X^{\bullet }:\operatorname{N}(\unicode[STIX]{x1D6E5})\rightarrow C$
is unital
$2$
-Segal.
Proof. Given a polygonal subdivision
$0\leqslant i\leqslant j\leqslant n$
, we augment the corresponding square (1.3) to the following diagram.
The
$1$
-Segal condition on
$X^{\bullet }$
implies that the left-hand square and outer square are pushouts. Hence, by [Reference LurieLur09, 4.4.2.1], the right-hand square is a pushout. Similarly, to obtain unitality, we augment the square (1.4) to the following diagram.
The 1-Segal condition on
$X^{\bullet }$
implies that the top and outer squares are pushouts so that, again by [Reference LurieLur09, 4.4.2.1], the bottom square is a pushout.◻
1.2 Cyclically ordered sets and ribbon graphs
Let
$S$
be a compact oriented surface, possibly with boundary
$\unicode[STIX]{x2202}S$
, together with a chosen finite subset
$M\subset S$
of marked points. We call
$(S,M)$
stable if:
-
(1) every connected component of
$S$
has at least one marked point; -
(2) every connected component of
$\unicode[STIX]{x2202}S$
has at least one marked point; -
(3) every connected component of
$S$
that is homeomorphic to the
$2$
-sphere has at least two marked points.
In this section, we develop a categorified state sum formalism based on a version of the well-known combinatorial description of stable oriented marked surfaces
$(S,M)$
in terms of ribbon graphs.
1.2.1 Cyclically ordered sets
In § 1.1, we enlarged the simplex category
$\unicode[STIX]{x1D6E5}$
to the equivalent category
$\boldsymbol{\unicode[STIX]{x1D6E5}}$
of finite non-empty linearly ordered sets. In analogy, we introduce the category of non-empty finite cyclically ordered sets
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
following [Reference Dyckerhoff and KapranovDK15], which plays the same role for the cyclic category
$\unicode[STIX]{x1D6EC}$
.
Let
$J$
be a finite non-empty set. We define a cyclic order on
$J$
to be a transitive action of the group
$\mathbb{Z}$
. Note that any such action induces a simply transitive action of the group
$\mathbb{Z}/N\mathbb{Z}$
on
$J$
where
$N$
denotes the cardinality of the set
$J$
.
Example 1.6. Let
$I$
be a finite non-empty linearly ordered set. We obtain a cyclic order on
$I$
as follows: let
$i_{0}<i_{1}<\cdots <i_{n}$
denote the elements of
$I$
. We set, for
$0\leqslant k<n$
,
$i_{k}+1=i_{k+1}$
, and
$i_{n}+1=i_{0}$
. We call the resulting cyclic order on
$I$
the cyclic closure of the given linear order. We denote the cyclic closure of the standard ordinal
$[n]$
by
$\langle n\rangle$
.
Example 1.7. More generally, let
$f:J\rightarrow J^{\prime }$
be a map of finite non-empty sets. Assume that
$J^{\prime }$
carries a cyclic order and that every fiber of
$f$
is equipped with a linear order. We define the lexicographic cyclic order on
$J$
as follows: let
$j\in J$
. If
$j$
is not maximal in its fiber then we define
$j+1$
to be the successor to
$j$
in its fiber. If
$j$
is maximal in its fiber, then we define
$j+1$
to be the minimal element of the successor fiber. (The cyclic order on
$J^{\prime }$
induces a cyclic order on the fibers of
$f$
where we simply skip empty fibers.)
A morphism
$J\rightarrow J^{\prime }$
of cyclically ordered sets consists of
-
(1) a map
$f:J\rightarrow J^{\prime }$
of underlying sets, -
(2) the choice of a linear order on every fiber of
$J^{\prime }$
such that the cyclic order on
$J$
is the lexicographic order from Example 1.7. We denote the resulting category of cyclically ordered sets by
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
. Given a cyclically ordered set
$J$
, we define the set of interstices
$$\begin{eqnarray}J^{\vee }=\operatorname{Hom}_{\boldsymbol{\unicode[STIX]{x1D6EC}}}(J,\langle 0\rangle ),\end{eqnarray}$$
which, by definition, is the set of linear orders on
$J$
whose cyclic closure agrees with the given cyclic order on
$J$
. If the set
$J$
has cardinality
$n+1$
, then we may identify
$J^{\vee }$
with the set of isomorphisms
$J\rightarrow \langle n\rangle$
in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
. The
$\mathbb{Z}$
-action on
$\langle n\rangle$
induces a
$\mathbb{Z}$
-action on
$J^{\vee }$
that defines a cyclic order.
Proposition 1.8. The association
$J\mapsto J^{\vee }$
extends to an equivalence of categories
$\boldsymbol{\unicode[STIX]{x1D6EC}}^{\operatorname{op}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}$
.
Proof. Given a morphism
$f:J\rightarrow J^{\prime }$
in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
, we have to define a dual
$f^{\vee }:(J^{\prime })^{\vee }\rightarrow J^{\vee }$
. The datum of
$f$
includes a choice of linear order on each fiber of
$f$
. Given a linear order on
$J^{\prime }$
, we can form the lexicographic linear order on
$J$
by a linear analog of the construction in Example 1.7. This defines the map
$f^{\vee }$
on underlying sets. We further have to define a linear order on the fibers of
$f^{\vee }$
. Given linear orders
$h:J^{\prime }\cong [n]$
and
$h^{\prime }:J^{\prime }\cong [n]$
such that
$f^{\vee }(h)=f^{\vee }(h^{\prime })$
, we fix any
$j\in J$
and declare
$h\leqslant h^{\prime }$
if
$h(j)\leqslant h^{\prime }(j)$
. This defines a linear order on each fiber of
$f^{\vee }$
which does in fact not depend on the chosen element
$j\in J$
. To verify that
$J\mapsto J^{\vee }$
is an equivalence, we observe that the double dual is naturally equivalent to the identity functor: an element
$j\in J$
determines a linear order on
$J^{\vee }$
by declaring, for
$h:J\cong [n]$
and
$h^{\prime }:J\cong [n]$
,
$h\leqslant h^{\prime }$
if
$h(j)\leqslant h^{\prime }(j)$
. We leave to the reader the verification that this association defines an isomorphism
$$\begin{eqnarray}J\rightarrow (J^{\vee })^{\vee }\end{eqnarray}$$
in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
that extends to a natural isomorphism between the identity functor and the double dual.◻
We refer to the equivalence
$\boldsymbol{\unicode[STIX]{x1D6EC}}^{\operatorname{op}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}$
as interstice duality. The following lemma will be important for the interplay between cyclic
$2$
-Segal objects and ribbon graphs.
Lemma 1.9. Let
$I,J$
be finite sets with elements
$i\in I$
,
$j\in J$
, and consider the pullback square
where the maps
$p$
and
$q$
are determined by
$p^{-1}(i)=\{i\}$
and
$q^{-1}(j)=\{j\}$
. Assume that
$K$
is non-empty and that the sets
$I$
and
$J$
are equipped with cyclic orders. Then the following hold.
-
(1) The above diagram lifts uniquely to a diagram in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
such that the induced cyclic orders in
$I$
and
$J$
are the given ones. -
(2) The resulting square in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
is a pullback square.
Proof. Follows by direct inspection.◻
Example 1.11. Consider the diagram of linearly ordered sets
corresponding to a polygonal subdivision
$0\leqslant i<j\leqslant n$
of a planar convex polygon as in § 1.1. Passing to cyclic closures we obtain a diagram in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
. By applying interstice duality we obtain a diagram in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
of the form (1.10) which is hence a pullback diagram. We deduce that the original diagram (1.12) is a pushout diagram in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
. Further, the same argument implies that, for every
$0\leqslant k<n$
, the diagram
is a pushout diagram in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
.
Definition 1.14. Let
$C$
be an
$\infty$
-category. A cocyclic object
$X:\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}})\rightarrow C$
is called (unital)
$2$
-Segal (respectively
$1$
-Segal) if the underlying cosimplicial object is (unital)
$2$
-Segal (respectively
$1$
-Segal).
Remark 1.15. Note, that the diagram
is a pushout diagram in
$\boldsymbol{\unicode[STIX]{x1D6E5}}$
and the
$1$
-Segal condition requires a cosimplicial object
$X:\unicode[STIX]{x1D6E5}\rightarrow C$
to preserve this pushout. In light of this observation, the
$2$
-Segal condition becomes very natural for cocyclic objects: while the cyclic closure of (1.16) is not a pushout square in
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
, the squares (1.12) and (1.13) are pushout squares. The
$2$
-Segal condition requires that these pushouts are preserved.
1.2.2 Ribbon graphs
A graph
$\unicode[STIX]{x1D6E4}$
is a pair of finite sets
$(H,V)$
equipped with an involution
$\unicode[STIX]{x1D70F}:H\rightarrow H$
and a map
$s:H\rightarrow V$
. The elements of the set
$H$
are called half-edges. We call a half-edge external if it is fixed by
$\unicode[STIX]{x1D70F}$
, and internal otherwise. A pair of
$\unicode[STIX]{x1D70F}$
-conjugate internal half-edges is called an edge and we denote the set of edges by
$E$
. The elements of
$V$
are called vertices. Given a vertex
$v$
, the half-edges in the set
$H(v)=s^{-1}(v)$
are said to be incident to
$v$
. A graph with one vertex and
$n$
half-edges equipped with the trivial involution is called an
$n$
-corolla.
A ribbon graph is a graph
$\unicode[STIX]{x1D6E4}$
where, for every vertex
$v$
of
$\unicode[STIX]{x1D6E4}$
, the set
$H(v)$
of half-edges incident to
$v$
is equipped with a cyclic order. We give an interpretation of this datum in terms of the category of cyclically ordered sets: let
$\unicode[STIX]{x1D6E4}$
be a graph. We define the incidence category
$I(\unicode[STIX]{x1D6E4})$
to have set of objects given by
$V\cup E$
and, for every internal half-edge
$h$
, a unique morphism from the vertex
$s(h)$
to the edge
$\{h,\unicode[STIX]{x1D70F}(h)\}$
. We define a functor
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}:I(\unicode[STIX]{x1D6E4})\longrightarrow {\mathcal{S}}et\end{eqnarray}$$
that, on objects, associates to a vertex
$v$
the set
$H(v)$
and to an edge
$e$
the set of half-edges underlying
$e$
. To a morphism
$v\rightarrow e$
, given by a half-edge
$h\in H(v)$
with
$h\in e$
, we associate the map
$\unicode[STIX]{x1D70B}:H(v)\rightarrow e$
which is determined by
$\unicode[STIX]{x1D70B}^{-1}(h)=\{h\}$
so that
$\unicode[STIX]{x1D70B}$
maps
$H(v)\setminus h$
to
$\unicode[STIX]{x1D70F}(h)$
. We call the functor
$\unicode[STIX]{x1D6FE}$
the incidence diagram of the graph
$\unicode[STIX]{x1D6E4}$
.
Proposition 1.17. Let
$\unicode[STIX]{x1D6E4}$
be a graph. A ribbon structure on
$\unicode[STIX]{x1D6E4}$
is equivalent to a lift
of the incidence diagram of
$\unicode[STIX]{x1D6E4}$
where
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
denotes the category of cyclically ordered sets.
The advantage of the interpretation of a ribbon structure given in Proposition 1.17 is that it facilitates the passage to interstices: given a ribbon graph
$\unicode[STIX]{x1D6E4}$
with corresponding incidence diagram
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}:I(\unicode[STIX]{x1D6E4})\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}},\end{eqnarray}$$
we introduce the coincidence diagram
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}:I(\unicode[STIX]{x1D6E4})^{\operatorname{op}}\longrightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}\end{eqnarray}$$
obtained by postcomposing
$\unicode[STIX]{x1D6FE}^{\operatorname{op}}$
with the interstice duality functor
$\boldsymbol{\unicode[STIX]{x1D6EC}}^{\operatorname{op}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}$
.
A morphism
$(f,\unicode[STIX]{x1D702}):\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$
of ribbon graphs consists of:
-
(1) a functor
$f:I(\unicode[STIX]{x1D6E4})\rightarrow I(\unicode[STIX]{x1D6E4}^{\prime })$
of incidence categories; -
(2) a natural transformation
$\unicode[STIX]{x1D702}:f^{\ast }\unicode[STIX]{x1D6FE}^{\prime }\rightarrow \unicode[STIX]{x1D6FE}$
of incidence diagrams.
We denote the resulting category of ribbon graphs by
${\mathcal{R}}ib$
.
Example 1.18. Let
$\unicode[STIX]{x1D6E4}$
be a graph and let
$e$
be an edge incident to two distinct vertices
$v$
and
$w$
. We define a new graph
$\unicode[STIX]{x1D6E4}^{\prime }$
obtained from
$\unicode[STIX]{x1D6E4}$
by contracting
$e$
as follows: the set of half-edges
$H^{\prime }$
is given by
$H\setminus e$
and the set of vertices
$V^{\prime }$
is obtained from
$V$
by identifying
$v$
and
$w$
. The involution
$\unicode[STIX]{x1D70F}$
on
$H$
restricts to an involution
$\unicode[STIX]{x1D70F}^{\prime }$
on
$H^{\prime }$
. We define
$s:H^{\prime }\rightarrow V^{\prime }$
as the composite of the restriction of
$s:H\rightarrow V$
to
$H^{\prime }$
and the quotient map
$V\rightarrow V^{\prime }$
.
We obtain a natural functor
$f:I(\unicode[STIX]{x1D6E4})\rightarrow I(\unicode[STIX]{x1D6E4}^{\prime })$
of incidence categories which collapses the objects
$v$
,
$w$
, and
$e$
to
$\overline{v}$
. Denoting by
$\unicode[STIX]{x1D6FE}$
and
$\unicode[STIX]{x1D6FE}^{\prime }$
the set-valued incidence diagrams, we construct a natural transformation
$\unicode[STIX]{x1D702}:f^{\ast }\unicode[STIX]{x1D6FE}^{\prime }\rightarrow \unicode[STIX]{x1D6FE}$
as follows. On objects of
$I(\unicode[STIX]{x1D6E4})$
different from
$v$
,
$w$
, and
$e$
, we define
$\unicode[STIX]{x1D702}$
to be the identity map. To obtain the values of
$\unicode[STIX]{x1D702}$
at
$v$
,
$w$
, and
$e$
, note that we have a natural commutative diagram
as in Lemma 1.9. The maps in the diagram determine the values of
$\unicode[STIX]{x1D702}$
as indicated.
Finally, assume that
$\unicode[STIX]{x1D6E4}$
carries a ribbon structure. Then Lemma 1.9(1) implies that
$\unicode[STIX]{x1D6E4}^{\prime }$
carries a unique ribbon structure such that the natural transformation
$\unicode[STIX]{x1D702}$
lifts to
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
-valued incidence diagrams.
A morphism of ribbon graphs
$\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$
as constructed in Example 1.18 is called an edge contraction. Note that the category
${\mathcal{R}}ib$
contains morphisms that cannot be obtained as compositions of edge contractions: for example, it contains a copy of the cyclic category
$\unicode[STIX]{x1D6EC}$
, given by the full subcategory spanned by the corollas. The following proposition indicates the relevance of Lemma 1.9(2).
Proposition 1.19. Let
$\unicode[STIX]{x1D6E4}$
be a ribbon graph and let
$(f,\unicode[STIX]{x1D702}):\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$
be an edge contraction. Then the natural transformation
$$\begin{eqnarray}\unicode[STIX]{x1D702}:f^{\ast }\unicode[STIX]{x1D6FE}^{\prime }\longrightarrow \unicode[STIX]{x1D6FE}\end{eqnarray}$$
exhibits
$\unicode[STIX]{x1D6FE}^{\prime }$
as a right Kan extension of
$\unicode[STIX]{x1D6FE}$
along
$f$
.
Proof. By the pointwise formula for right Kan extensions, it suffices to verify that, for every object
$y\in I(\unicode[STIX]{x1D6E4}^{\prime })$
, the natural transformation
$\unicode[STIX]{x1D702}$
exhibits
$\unicode[STIX]{x1D6FE}^{\prime }(y)$
as the limit of the diagram
$$\begin{eqnarray}y/f\longrightarrow \unicode[STIX]{x1D6EC},\quad (x,y\rightarrow f(x))\mapsto \unicode[STIX]{x1D6FE}(x).\end{eqnarray}$$
Unravelling the definitions, this is a trivial condition unless
$y$
is the object corresponding to the vertex
$\overline{v}$
under the contracted edge. For
$y=\overline{v}$
the condition reduces to Lemma 1.9(2).◻
1.2.3 State sums on ribbon graphs
We introduce a category
${\mathcal{R}}ib^{\ast }$
with objects given by pairs
$(\unicode[STIX]{x1D6E4},x)$
where
$\unicode[STIX]{x1D6E4}$
is a ribbon graph and
$x$
is an object of the incidence category
$I(\unicode[STIX]{x1D6E4})$
. A morphism
$(\unicode[STIX]{x1D6E4},x)\rightarrow (\unicode[STIX]{x1D6E4}^{\prime },y)$
consists of a morphism
$(f,\unicode[STIX]{x1D702}):\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$
of ribbon graphs together with a morphism
$y\rightarrow f(x)$
in
$I(\unicode[STIX]{x1D6E4}^{\prime })$
. The category
${\mathcal{R}}ib^{\ast }$
comes equipped with a forgetful functor
$\unicode[STIX]{x1D70B}:{\mathcal{R}}ib^{\ast }\rightarrow {\mathcal{R}}ib$
and an evaluation functor
$$\begin{eqnarray}\operatorname{ev}:{\mathcal{R}}ib^{\ast }\longrightarrow \boldsymbol{\unicode[STIX]{x1D6EC}},\quad (\unicode[STIX]{x1D6E4},x)\mapsto \unicode[STIX]{x1D6FF}(x),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FF}$
denotes the coincidence diagram of
$\unicode[STIX]{x1D6E4}$
. Let
$C$
be an
$\infty$
-category with colimits, and let
$X:\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}})\rightarrow C$
be a cocyclic object in
$C$
. The functor
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X}=\operatorname{N}(\unicode[STIX]{x1D70B})_{!}(X\circ \operatorname{N}(\operatorname{ev})):\operatorname{N}({\mathcal{R}}ib)\longrightarrow C\end{eqnarray}$$
is called the state sum functor of
$X$
. Here,
$\operatorname{N}(\unicode[STIX]{x1D70B})_{!}$
denotes the
$\infty$
-categorical left Kan extension defined in [Reference LurieLur09, 4.3.3.2]. For a ribbon graph
$\unicode[STIX]{x1D6E4}$
, the object
$X(\unicode[STIX]{x1D6E4}):=\unicode[STIX]{x1D70C}_{X}(\unicode[STIX]{x1D6E4})$
is called the state sum of
$X$
on
$\unicode[STIX]{x1D6E4}$
.
Proposition 1.20. The state sum of
$X$
on
$\unicode[STIX]{x1D6E4}$
admits the formula
$$\begin{eqnarray}X(\unicode[STIX]{x1D6E4})\simeq \operatorname{colim}X\circ \operatorname{N}(\unicode[STIX]{x1D6FF}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FF}:I(\unicode[STIX]{x1D6E4})^{\operatorname{op}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}$
denotes the coincidence diagram of
$\unicode[STIX]{x1D6E4}$
.
Proof. By the pointwise formula for left Kan extensions, we have
$$\begin{eqnarray}X(\unicode[STIX]{x1D6E4})=\operatorname{colim}X\circ \operatorname{N}(\operatorname{ev})_{|\operatorname{N}(\unicode[STIX]{x1D70B}/\unicode[STIX]{x1D6E4})}.\end{eqnarray}$$
The nerve of the functor
$$\begin{eqnarray}I(\unicode[STIX]{x1D6E4})^{\operatorname{op}}\longrightarrow \unicode[STIX]{x1D70B}/\unicode[STIX]{x1D6E4},\quad x\mapsto ((\unicode[STIX]{x1D6E4},x),\unicode[STIX]{x1D6E4}\overset{\operatorname{id}}{\rightarrow }\unicode[STIX]{x1D6E4})\end{eqnarray}$$
is cofinal, which implies the claim.◻
Example 1.21. The universal example of a cocyclic object with values in an
$\infty$
-category with colimits is the Yoneda embedding
$j:\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}})\rightarrow P(\boldsymbol{\unicode[STIX]{x1D6EC}})$
where
$P(\boldsymbol{\unicode[STIX]{x1D6EC}})$
denotes the
$\infty$
-category
$\operatorname{Fun}(\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}})^{\operatorname{op}},S)$
of cyclic spaces. We obtain a functor
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{j}:\operatorname{N}({\mathcal{R}}ib)\longrightarrow P(\boldsymbol{\unicode[STIX]{x1D6EC}})\end{eqnarray}$$
that realizes a ribbon graph as a cyclic space. This functor is the universal state sum: given any cocyclic object
$X:\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}})\rightarrow C$
where
$C$
has colimits, we have
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X}\simeq j_{!}X\circ \unicode[STIX]{x1D70C}_{j},\end{eqnarray}$$
where we use Proposition 1.20 and the fact [Reference LurieLur09, 5.1.5.5] that
$j_{!}X$
commutes with colimits. We will use the notation
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D6E4}):=\unicode[STIX]{x1D70C}_{j}(\unicode[STIX]{x1D6E4})\end{eqnarray}$$
for the state sum of
$j$
on
$\unicode[STIX]{x1D6E4}$
.
The following proposition explains the relevance of the
$2$
-Segal condition for state sums.
Proposition 1.23. Let
$C$
be an
$\infty$
-category with colimits and let
$X:\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}})\rightarrow C$
be a cocyclic object. Then
$X$
is unital
$2$
-Segal if and only if the state sum functor
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X}:\operatorname{N}({\mathcal{R}}ib)\longrightarrow C,\quad \unicode[STIX]{x1D6E4}\mapsto X(\unicode[STIX]{x1D6E4})\end{eqnarray}$$
maps edge contractions in
${\mathcal{R}}ib$
to equivalences in
$C$
.
Proof. Let
$(f,\unicode[STIX]{x1D702}):\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$
be a morphism of ribbon graphs. The associated morphism
$\unicode[STIX]{x1D70C}_{X}(f,\unicode[STIX]{x1D702}):X(\unicode[STIX]{x1D6E4})\rightarrow X(\unicode[STIX]{x1D6E4}^{\prime })$
is given by the composite
$$\begin{eqnarray}\operatorname{colim}(X\circ \unicode[STIX]{x1D6FF})\overset{X\circ \unicode[STIX]{x1D702}^{\vee }}{\longrightarrow }\operatorname{colim}(X\circ \unicode[STIX]{x1D6FF}^{\prime }\circ f^{\operatorname{op}})\longrightarrow \operatorname{colim}(X\circ \unicode[STIX]{x1D6FF}^{\prime }).\end{eqnarray}$$
We claim that, if
$(f,\unicode[STIX]{x1D702})$
is an edge contraction, then
$X\circ \unicode[STIX]{x1D702}^{\vee }$
exhibits
$X\circ \unicode[STIX]{x1D6FF}^{\prime }$
as a left Kan extension of
$X\circ \unicode[STIX]{x1D6FF}$
. This implies the result since a colimit is given by the left Kan extension to the final category and left Kan extension functors are functorial in the sense
$f_{!}\circ g_{!}\simeq (f\circ g)_{!}$
. The claim follows immediately from the argument of Proposition 1.19, Lemma 1.9, and Remark 1.15.◻
Remark 1.24. In the situation of Proposition 1.23, we may restrict ourselves to the subcategory
${\mathcal{R}}ib^{\prime }\subset {\mathcal{R}}ib$
generated by edge contractions and isomorphisms. Then by the statement of the theorem, we obtain a functor
$$\begin{eqnarray}\operatorname{N}({\mathcal{R}}ib^{\prime })_{\simeq }\rightarrow C,\quad \unicode[STIX]{x1D6E4}\mapsto X(\unicode[STIX]{x1D6E4}),\end{eqnarray}$$
where
$\operatorname{N}({\mathcal{R}}ib^{\prime })_{\simeq }=\operatorname{Sing}|\operatorname{N}({\mathcal{R}}ib^{\prime })|$
denotes the
$\infty$
-groupoid completion of the
$\infty$
-category
$\operatorname{N}({\mathcal{R}}ib^{\prime })$
. The automorphism group in
$\operatorname{N}({\mathcal{R}}ib^{\prime })_{\simeq }$
of a ribbon graph
$\unicode[STIX]{x1D6E4}$
that represents a stable oriented marked surface
$(S,M)$
can be identified with the mapping class group
$\operatorname{Mod}(S,M)$
. The above functor implies the existence of an
$\infty$
-categorical action of
$\operatorname{Mod}(S,M)$
on
$X(\unicode[STIX]{x1D6E4})$
, which is a main result of [Reference Dyckerhoff and KapranovDK13].
1.3 Paracyclically ordered sets and framed graphs
Let
$(S,M)$
be a stable oriented marked surface. We may interpret the orientation as a reduction of the structure group of the tangent bundle of
$S\setminus M$
along
$\operatorname{GL}_{2}^{+}(\mathbb{R})\subset \operatorname{GL}_{2}(\mathbb{R})$
. We define a framing of
$(S,M)$
as a further lift of the structure group along the universal cover
$$\begin{eqnarray}\widetilde{\operatorname{GL}_{2}^{+}(\mathbb{R})}\longrightarrow \operatorname{GL}_{2}^{+}(\mathbb{R}).\end{eqnarray}$$
Up to contractible choice, this datum is equivalent to a trivialization of the tangent bundle of
$S\setminus M$
. In this section, we describe a state sum formalism based on a combinatorial model for stable framed marked surfaces as developed in [Reference Dyckerhoff and KapranovDK15]. This amounts to a variation of the constructions in the previous section, obtained by replacing the cyclic category by the paracyclic category.
1.3.1 Paracyclically ordered sets
Let
$J$
be a finite non-empty set. We define a paracyclic order on
$J$
to be a cyclic order on
$J$
together with the choice of a
$\mathbb{Z}$
-torsor
$\widetilde{J}$
and a
$\mathbb{Z}$
-equivariant map
$\widetilde{J}\rightarrow J$
. A morphism of paracyclically ordered sets
$(J,\widetilde{J})\rightarrow (J^{\prime },\widetilde{J^{\prime }})$
consists of a commutative diagram of sets
such that
$\widetilde{f}$
is monotone with respect to the
$\mathbb{Z}$
-torsor linear orders. The lift
$\widetilde{f}$
equips
$f$
naturally with the structure of a morphism of cyclically ordered sets so that we obtain a forgetful functor
$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }\longrightarrow \boldsymbol{\unicode[STIX]{x1D6EC}},\end{eqnarray}$$
where
$\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }$
denotes the category of paracyclically ordered sets. As for cyclically ordered sets, there is a skeleton
$\unicode[STIX]{x1D6EC}_{\infty }\subset \boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }$
consisting of standard paracyclically ordered sets
$(\langle n\rangle ,\widetilde{\langle n\rangle })$
where we define
$\widetilde{\langle n\rangle }=\mathbb{Z}$
and
$\widetilde{\langle n\rangle }\rightarrow \langle n\rangle$
is given by the natural quotient map.
Given a paracyclically ordered set
$(J,\widetilde{J})$
, then the cyclic order on the interstice dual
$J^{\vee }$
lifts to a natural paracyclic order given by
$$\begin{eqnarray}\widetilde{J^{\vee }}=\operatorname{Hom}_{\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }}((J,\widetilde{J}),(\langle 0\rangle ,\widetilde{\langle 0\rangle })).\end{eqnarray}$$
This construction extends the self duality of
$\boldsymbol{\unicode[STIX]{x1D6EC}}$
to one for
$\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }$
. All statements of § 1.2.1 hold mutatis mutandis for
$\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }$
.
1.3.2 Framed graphs and state sums
We define a framed graph
$\unicode[STIX]{x1D6E4}$
to be a graph
$\unicode[STIX]{x1D6E4}$
equipped with a lift
of the incidence diagram of
$\unicode[STIX]{x1D6E4}$
where
$\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }$
denotes the category of paracyclically ordered sets. Framed graphs form a category
${\mathcal{R}}ib_{\infty }$
which is defined in complete analogy with
${\mathcal{R}}ib$
.
Let
$\unicode[STIX]{x1D6E4}$
be a framed graph with incidence diagram
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}:I(\unicode[STIX]{x1D6E4})\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }.\end{eqnarray}$$
Let
$e=\{h,\unicode[STIX]{x1D70F}(h)\}$
be an edge in
$\unicode[STIX]{x1D6E4}$
incident to the vertices
$v=s(h)$
and
$w=s(\unicode[STIX]{x1D70F}(h))$
, and let
$h^{\prime }$
be a half-edge incident to
$w$
. Let
$\widetilde{h}$
be a lift of
$h$
to an element of the
$\mathbb{Z}$
-torsor
$\widetilde{H(v)}$
that is part of the paracyclic structure on
$H(v)$
. Then we may transport this lift along the edge
$e$
to obtain a lift of
$h^{\prime }$
to an element
$\widetilde{h^{\prime }}$
of
$\widetilde{H(w^{\prime })}$
as follows: there is a unique lift of
$\widetilde{\unicode[STIX]{x1D70F}(h)}\in \widetilde{H(w^{\prime })}$
of
$\unicode[STIX]{x1D70F}(h)$
which maps to
$\widetilde{h}-1$
under
$\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F}(h))$
. We then set
$\widetilde{h^{\prime }}=\widetilde{\unicode[STIX]{x1D70F}(h)}+i$
where
$i\geqslant 0$
is minimal such that
$\widetilde{h^{\prime }}$
lifts
$h^{\prime }$
. Iterating this transport along a loop
$l$
returning to the half-edge
$h$
, we obtain another lift
$\overline{h}$
of
$h$
. The integer
$\overline{h}-\widetilde{h}$
only depends on
$l$
and we refer to it as the winding number of l.
Remark 1.25. As explained in [Reference Dyckerhoff and KapranovDK15], framed graphs provide a combinatorial model for stable marked surfaces
$(S,M)$
equipped with a trivialization of the tangent bundle of
$S\setminus M$
. The above combinatorial construction then coincides with the geometric concept of winding number computed with respect to the framing.
Let
$C$
be an
$\infty$
-category with colimits, and let
$X^{\bullet }:\operatorname{N}(\boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty })\rightarrow C$
be a coparacyclic object in
$C$
. Then, given a framed graph
$\unicode[STIX]{x1D6E4}$
, we have a state sum
$$\begin{eqnarray}X(\unicode[STIX]{x1D6E4})=\operatorname{colim}X\circ \unicode[STIX]{x1D6FF},\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FF}:I(\unicode[STIX]{x1D6E4})^{\operatorname{op}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6EC}}_{\infty }$
denotes the coincidence diagram of
$\unicode[STIX]{x1D6E4}$
. The various state sums naturally organize into a functor
$$\begin{eqnarray}\operatorname{N}({\mathcal{R}}ib_{\infty })\longrightarrow C,\quad \unicode[STIX]{x1D6E4}\mapsto X(\unicode[STIX]{x1D6E4}).\end{eqnarray}$$
Remark 1.26. As shown in [Reference Dyckerhoff and KapranovDK15], the state sum
$X(\unicode[STIX]{x1D6E4})$
of a framed graph with values in a coparacyclic unital
$2$
-Segal object
$X$
admits an action of the framed mapping class group of the surface.
1.4 The universal loop space
We give a first example of a state sum that will play an important role later on. Consider the functor
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}\longrightarrow \operatorname{Grp},\quad \langle n\rangle \mapsto \unicode[STIX]{x1D70B}_{1}(D/\{0,1,\ldots ,n\}),\end{eqnarray}$$
where
$D/\{0,1,\ldots ,n\}$
denotes the quotient of the unit disk by
$n+1$
marked points on the boundary. Replacing the fundamental groups by their nerves, we obtain a functor
$$\begin{eqnarray}L^{\bullet }:\operatorname{N}(\unicode[STIX]{x1D6EC})\longrightarrow S_{\ast },\end{eqnarray}$$
where
$S_{\ast }$
denotes the
$\infty$
-category of pointed spaces. The cosimplicial pointed space underlying
$L^{\bullet }$
is
$1$
-Segal and hence, by Proposition 1.5, unital
$2$
-Segal.
Remark 1.28. Note that the group
$\unicode[STIX]{x1D70B}_{1}(D/\{0,1,\ldots ,n\})$
is a free group on
$n$
generators so that
$L^{n}$
is equivalent to a bouquet of
$n$
one-dimensional spheres. Given any pointed space
$X$
, the cyclic pointed
$1$
-Segal space
$\operatorname{Map}(L^{\bullet },X)$
describes the loop space
$\unicode[STIX]{x1D6FA}X$
together with its natural group structure. Similarly, the cocyclic pointed
$1$
-Segal space
$L^{\bullet }\otimes X$
describes the suspension of
$X$
together with its natural cogroup structure.
Proposition 1.29. Let
$(S,M)$
be a stable marked oriented surface represented by a ribbon graph
$\unicode[STIX]{x1D6E4}$
. Then we have
$$\begin{eqnarray}L(\unicode[STIX]{x1D6E4})\simeq S/M.\end{eqnarray}$$
Proof. We can compute the state sum defining
$L(\unicode[STIX]{x1D6E4})$
explicitly as a homotopy colimit in the category of pointed spaces. We may replace the diagram by a homotopy equivalent one which assigns to each
$(n+1)$
-corolla of the graph
$\unicode[STIX]{x1D6E4}$
the space
$D/\{0,1,\ldots ,n\}$
. The homotopy colimit is then obtained by identifying the boundary cycles of the various spaces
$D/\{0,1,\ldots ,n\}$
according to the incidence relations given by the edges of the ribbon graphs. It is apparent that the resulting space is equivalent to the quotient space
$S/M$
.◻
Example 1.30. As an illustration of the gluing procedure described in the proof of Proposition 1.29, consider the ribbon graph with one vertex and two half-edges forming a loop at the vertex. This graph models a sphere
$S^{2}$
with two marked points
$M$
. The colimit in the proof describes the space
$S^{2}/M$
as obtained from
$D/\{0,1\}$
by identifying the two boundary circles given by the images of the two boundary arcs connecting the points
$0$
and
$1$
on
$\unicode[STIX]{x2202}D$
.
For a slight elaboration on Proposition 1.29, let
$C$
be a stable
$\infty$
-category with colimits and let
$E$
be an object of
$C$
. By [Reference LurieLur11, 1.4.2.21], the functor
$$\begin{eqnarray}\operatorname{Sp}(C)=\operatorname{Fun}^{c}(S_{\ast }^{\operatorname{fin}},C)\rightarrow C,\quad f\mapsto f(S^{0})\end{eqnarray}$$
from the
$\infty$
-category of spectrum objects in
$C$
to
$C$
is an equivalence. Therefore, the object
$E$
defines an essentially unique functor
$$\begin{eqnarray}E:S_{\ast }^{\operatorname{fin}}\rightarrow C,\end{eqnarray}$$
which we still denote by
$E$
.
Definition 1.32. Let
$(X,Y)$
be a pair of finite spaces. We introduce the pointed quotient space
$X/Y$
as the pushout
in
$S$
. The object
$E(X/Y)$
of
$C$
is called the relative homology of the pair
$(X,Y)$
with coefficients in
$E$
. In the case when
$C$
is the category of spectra, this terminology agrees with the customary one.
We introduce the cocyclic object
$$\begin{eqnarray}L_{E}=E(L^{\bullet }):\operatorname{N}(\unicode[STIX]{x1D6EC})\longrightarrow C\end{eqnarray}$$
obtained from (1.27) by postcomposing with
$E:S_{\ast }^{\operatorname{fin}}\rightarrow C$
.
Proposition 1.34. Let
$(S,M)$
be a stable marked oriented surface represented by a ribbon graph
$\unicode[STIX]{x1D6E4}$
. Then we have an equivalence
$$\begin{eqnarray}L_{E}(\unicode[STIX]{x1D6E4})\simeq E(S/M)\end{eqnarray}$$
in
$C$
.
Proof. The functor
$E(-)$
commutes with finite colimits so that the statement follows immediately from Proposition 1.29.◻
Remark 1.35. We may pullback a cocyclic
$2$
-Segal object
$X^{\bullet }$
along the functor
$\unicode[STIX]{x1D6EC}_{\infty }\rightarrow \unicode[STIX]{x1D6EC}$
to obtain a coparacyclic
$2$
-Segal object
$\widetilde{X^{\bullet }}$
. Given a framed graph
$\unicode[STIX]{x1D6E4}$
, we have
$$\begin{eqnarray}\widetilde{X}(\unicode[STIX]{x1D6E4})\simeq X(\overline{\unicode[STIX]{x1D6E4}}),\end{eqnarray}$$
where
$\overline{\unicode[STIX]{x1D6E4}}$
is the ribbon graph underlying
$\unicode[STIX]{x1D6E4}$
. In particular, in the context of Proposition 1.34, we obtain
$$\begin{eqnarray}\widetilde{L_{E}}(\unicode[STIX]{x1D6E4})\simeq E(S/M).\end{eqnarray}$$
2 Differential graded categories
2.1 Morita equivalences
We introduce some terminology for the derived Morita theory of differential graded categories and refer the reader to [Reference TabuadaTab05, Reference ToënToë07] for more detailed treatments. Let
$k$
be a commutative ring, and let
$Cat_{\operatorname{dg}}$
be the category of small
$k$
-linear differential
$\mathbb{Z}$
-graded categories. Recall that a functor
$f:A\rightarrow B$
is called a quasi-equivalence if:
-
(1) the functor
$H^{0}(f):H^{0}(A)\rightarrow H^{0}(B)$
of homotopy categories is an equivalence of categories; -
(2) for every pair of objects
$(x,y)$
in
$A$
, the morphism
$f:\operatorname{Hom}_{A}(x,y)\rightarrow \operatorname{Hom}_{B}(f(x),f(y))$
of complexes is a quasi-isomorphism.
We denote by
$\operatorname{L}_{\operatorname{qe}}(Cat_{\operatorname{dg}})$
the
$\infty$
-category obtained by localizing
$Cat_{\operatorname{dg}}$
along quasi-equivalences [Reference LurieLur11, 1.3.4.1]. The collection of quasi-equivalences can be supplemented to a combinatorial model structure on
$Cat_{\operatorname{dg}}$
which facilitates calculations in
$\operatorname{L}_{\operatorname{qe}}(Cat_{\operatorname{dg}})$
.
Given dg categories
$A$
,
$B$
, we denote the dg category of enriched functors from
$A$
to
$B$
by
$\text{}\underline{\operatorname{Hom}}(A,B)$
. We denote by
$\operatorname{Mod}_{k}$
the dg category of unbounded complexes of
$k$
-modules, and further, by
$\operatorname{Mod}_{A}$
the dg category
$\text{}\underline{\operatorname{Hom}}(A^{\operatorname{op}},\operatorname{Mod}_{k})$
. We equip
$\operatorname{Mod}_{A}$
with the projective model structure and denote by
$\operatorname{Perf}_{A}\subset \operatorname{Mod}_{A}$
the full dg category spanned by those objects
$x$
such that:
-
(1)
$x$
is cofibrant; -
(2) the image of
$x$
in
$H^{0}(\operatorname{Mod}_{A})$
is compact, i.e.
$\operatorname{Hom}(x,-)$
commutes with coproducts.
Given a dg functor
$f:A\rightarrow B$
, we have a Quillen adjunction
$$\begin{eqnarray}f_{!}:\operatorname{Mod}_{A}\longrightarrow \operatorname{Mod}_{B}:f^{\ast }\end{eqnarray}$$
and obtain an induced functor
$$\begin{eqnarray}f_{!}:\operatorname{Perf}_{A}\longrightarrow \operatorname{Perf}_{B}\!.\end{eqnarray}$$
The functor
$f:A\rightarrow B$
is called a Morita equivalence if the induced functor (2.1) is a quasi-equivalence. We denote by
$\operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
the
$\infty$
-category obtained by localizing
$Cat_{\operatorname{dg}}$
along Morita equivalences. We have an adjunction
$$\begin{eqnarray}l:\operatorname{L}_{\operatorname{qe}}(Cat_{\operatorname{dg}})\longleftrightarrow \operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}}):i,\end{eqnarray}$$
where
$i$
is fully faithful so that
$l$
is a localization functor.
Let
$Cat_{\operatorname{dg}}^{(2)}$
denote the category of small
$\mathbf{k}$
-linear differential
$\mathbb{Z}/2\mathbb{Z}$
-graded categories. All of the above theory can be translated mutatis mutandis via the adjunction
$$\begin{eqnarray}P:Cat_{\operatorname{dg}}\longleftrightarrow Cat_{\operatorname{dg}}^{(2)}:Q,\end{eqnarray}$$
which is a Quillen adjunction with respect to an adaptation of the quasi-equivalence model structure on
$Cat_{\operatorname{dg}}^{(2)}$
. The periodization functor
$P$
associates to a differential
$\mathbb{Z}$
-graded category the differential
$\mathbb{Z}/2\mathbb{Z}$
-graded category with the same objects and
$\mathbb{Z}/2\mathbb{Z}$
-graded mapping complexes obtained by summing over all even (respectively odd) terms of the
$\mathbb{Z}$
-graded mapping complexes. We will refer to the
$\mathbb{Z}/2\mathbb{Z}$
-graded analogs of the above constructions via the superscript
$(2)$
.
Remark 2.3. Note that, due to the adjunction (2.2), the functor
$l$
commutes with colimits so that we may compute colimits in the category
$\operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
as colimits in
$\operatorname{L}_{\operatorname{qe}}(Cat_{\operatorname{dg}})$
. The latter category is equipped with the quasi-equivalence model structure so that, by [Reference LurieLur09], we can compute colimits as homotopy colimits with respect to this model structure. The analogous statement holds for the
$\mathbb{Z}/2\mathbb{Z}$
-graded variants.
2.2 Exact sequences of dg categories
A morphism in
$\operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
is called quasi-fully faithful if it is equivalent to the image of a quasi-fully faithful morphism under the localization functor
$\operatorname{N}(Cat_{\operatorname{dg}})\rightarrow \operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
. A pushout square
in
$\operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
with
$g$
quasi-fully faithful is called an exact sequence. The following technical statement will be used below.
Lemma 2.4. Quasi-fully faithful morphisms are stable under pushouts in
$\operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
.
Proof. The adjunction (2.2) implies that the left adjoint
$l:\operatorname{L}_{\operatorname{qe}}(Cat_{\operatorname{dg}})\rightarrow \operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}})$
preserves colimits so that it suffices to prove the corresponding statement for
$\operatorname{L}_{\operatorname{qe}}(Cat_{\operatorname{dg}})$
. To this end it suffices to show that quasi-fully faithful functors in the category
$Cat_{\operatorname{dg}}$
are stable under homotopy pushouts with respect to the quasi-equivalence model structure defined in [Reference TabuadaTab05]. Given a diagram
with
$g$
quasi-fully faithful, we may assume that all objects are cofibrant, and
$f$
,
$g$
are cofibrations so that the homotopy pushout is given by an ordinary pushout. Denoting by
$I$
the set of generating cofibrations of
$Cat_{\operatorname{dg}}$
, we may, by Quillen’s small object argument, factor the morphism
$f$
as
$$\begin{eqnarray}S\overset{f_{1}}{\rightarrow }\widetilde{S^{\prime }}\overset{f_{2}}{\rightarrow }S^{\prime },\end{eqnarray}$$
where
$f_{1}$
is a relative
$I$
-cell complex and
$f_{2}$
is a trivial fibration. Forming pushouts we obtain a diagram
Since
$\widetilde{g}$
is a cofibration, the bottom square is a homotopy pushout square and thus
$r$
is a quasi-equivalence. It hence suffices to show that
$\widetilde{g}$
is quasi-fully faithful so that we may assume that
$f$
is a relative
$I$
-cell complex. Using that filtered colimits of complexes are homotopy colimits [Reference Toën and VaquiéTV07], and hence preserve quasi-isomorphisms, we reduce to the case that
$f$
is the pushout of a single generating cofibration from
$I$
. This leaves us with the following two cases.
-
(1)
$S^{\prime }$
is obtained from
$S$
by adjoining one object, and the pushout
$T^{\prime }$
of (2.5) is obtained from
$T$
by adjoining one object. Clearly, the functor
$S^{\prime }\rightarrow T^{\prime }$
is quasi-fully faithful. -
(2)
$S^{\prime }$
is obtained from
$S$
by freely adjoining a morphism
$p:a\rightarrow b$
of some degree
$n$
between objects
$a,b$
of
$S$
where
$d(p)$
is a prescribed morphism
$q$
of
$S$
. The morphism complex between objects
$x,y$
in
$S^{\prime }$
can be described explicitly as where
$$\begin{eqnarray}S^{\prime }(x,y)=\bigoplus _{n\geqslant 0}S(x,a)\otimes kp\otimes S(b,a)\otimes kp\otimes \cdots \otimes S(b,y),\end{eqnarray}$$
$n$
copies of
$kp$
appear in the
$n$
th summand. The differential is given by the Leibniz rule where, upon replacing
$p$
by
$d(p)=q$
, we also compose with the neighboring morphisms so that the level is decreased from
$n$
to
$n-1$
. The morphism complexes of the pushout
$T^{\prime }$
admit an analogous expression with
$p$
replaced by
$g(p)$
. We have to show that, for every pair of objects, the morphism of complexes is a quasi-isomorphism. To this end, we filter both complexes by the level
$$\begin{eqnarray}S^{\prime }(x,y)\rightarrow T^{\prime }(g(x),g(y))\end{eqnarray}$$
$n$
. On the associated graded complexes we have a quasi-isomorphism, since
$g$
is quasi-fully faithful. The corresponding spectral sequence converges, which yields the desired quasi-isomorphism.
◻
Remark 2.7. The proof of the lemma works verbatim for
$Cat_{\operatorname{dg}}^{(2)}$
instead of
$Cat_{\operatorname{dg}}$
.
3 Topological Fukaya categories
3.1
$\mathbb{Z}/2\mathbb{Z}$
-graded
Let
$k$
be a commutative ring, and let
$R=k[z]$
denote the polynomial ring with coefficients in
$k$
, considered as a
$\mathbb{Z}/(n+1)$
-graded
$k$
-algebra with
$|z|=1$
. A matrix factorization
$(X,d_{X})$
of
$w=z^{n+1}$
consists of:
-
– a pair
$X^{0}$
,
$X^{1}$
of
$\mathbb{Z}/(n+1)$
-graded
$R$
-modules; -
– a pair of homogeneous
$R$
-linear homomorphismsof degree
$0$
;
such that
-
–
$d^{1}\circ d^{0}=w\operatorname{id}_{X^{0}}$
and
$d^{0}\circ d^{1}=w\operatorname{id}_{X^{1}}$
.
Example 3.1. For
$i,j\in \mathbb{Z}/(n+1)$
,
$i\neq j$
, we have a corresponding scalar matrix factorization
$[i,j]$
defined as
where the exponents of
$z$
are to be interpreted via their representatives in
$\{1,2,\ldots ,n\}$
. For
$i=j$
, we have two scalar matrix factorizations
and
which we denote by
$[i,i]_{r}$
and
$[i,i]_{l}$
, respectively.
Given matrix factorizations
$X$
,
$Y$
of
$w$
, we form the
$\mathbb{Z}/2\mathbb{Z}$
-graded
$k$
-module
$\operatorname{Hom}^{\bullet }(X,Y)$
with
$$\begin{eqnarray}\displaystyle \operatorname{Hom}^{0}(X,Y) & = & \displaystyle \operatorname{Hom}_{R}(X^{0},Y^{0})\oplus \operatorname{Hom}_{R}(X^{1},Y^{1}),\nonumber\\ \displaystyle \operatorname{Hom}^{1}(X,Y) & = & \displaystyle \operatorname{Hom}_{R}(X^{0},Y^{1})\oplus \operatorname{Hom}_{R}(X^{1},Y^{0}),\nonumber\end{eqnarray}$$
where
$\operatorname{Hom}_{R}$
denotes homogeneous
$R$
-linear homomorphisms of degree
$0$
. It is readily verified that the formula
$$\begin{eqnarray}d(f)=d_{Y}\circ f-(-1)^{|f|}f\circ d_{X}\end{eqnarray}$$
defines a differential on
$\operatorname{Hom}^{\bullet }(X,Y)$
, i.e.
$d^{2}=0$
. Therefore, the collection of all matrix factorizations of
$w$
organizes into a differential
$\mathbb{Z}/2\mathbb{Z}$
-graded
$k$
-linear category which we denote by
$\operatorname{MF}^{\mathbb{Z}/(n+1)}(k[z],z^{n+1})$
. We further define
$$\begin{eqnarray}\bar{\operatorname{F}}^{n}\subset \operatorname{MF}^{\mathbb{Z}/(n+1)}(k[z],z^{n+1})\end{eqnarray}$$
to be the full dg subcategory spanned by the scalar matrix factorizations from Example 3.1.
Theorem 3.2. The association
$n\mapsto \bar{\operatorname{F}}^{n}$
extends to a cocyclic object
$$\begin{eqnarray}\bar{\operatorname{F}}:\operatorname{N}(\unicode[STIX]{x1D6EC})\rightarrow \operatorname{L}_{\operatorname{mo}}(Cat_{\operatorname{dg}}^{(2)})\end{eqnarray}$$
which is unital 2-Segal.